A Second Raabe's Test and Other Series Tests

The classical D'Alembert's Ratio Test is a powerful test that we learn from calculus to determine convergence for a series of positive terms. Its range of applicability and ease of computation make this test extremely appealing. However, when the limiting ratio of the terms equals 1, then the test is inconclusive. Several series tests like Raabe's and Gauss' Tests have been proposed in order to address this case. These tests were also generalized by Kummer through Kummer's Test. More recently, a Second Ratio Test was constructed that also possessed an inconclusive case. This article presents a survey of existing series tests, introduces an extension of Raabe's Test to the Second Ratio Test, and proposes extensions of classical tests such as Gauss's Test and Kummer's Test. It also offers proofs of the aforementioned tests and a brief application of the Second Raabe's Test.